Mathematica® for Theoretical Physics ® Mathematica for Theoretical Physics Classical Mechanics and Nonlinear Dynamics Second Edition Gerd Baumann CD-ROM Included Gerd Baumann Department of Mathematics German University in Cairo GUC New Cairo City Main Entrance of Al Tagamoa Al Khames Egypt Gerd.Baumann@GUC.edu.eg This is a translated, expanded, and updated version of the original German version of the work “Mathematica® in der Theoretischen Physik,” published by Springer-Verlag Heidelberg, 1993 © Library of Congress Cataloging-in-Publication Data Baumann, Gerd [Mathematica in der theoretischen Physik English] Mathematica for theoretical physics / by Gerd Baumann.—2nd ed p cm Includes bibliographical references and index Contents: Classical mechanics and nonlinear dynamics — Electrodynamics, quantum mechanics, general relativity, and fractals ISBN 0-387-01674-0 Mathematical physics—Data processing Mathematica (Computer file) I Title QC20.7.E4B3813 2004 530′.285′53—dc22 ISBN-10: 0-387-01674-0 ISBN-13: 978-0387-01674-0 2004046861 e-ISBN 0-387-25113-8 Printed on acid-free paper © 2005 Springer Science+Business Media, Inc All rights reserved This work may not be translated or copied in whole or in part without the written permission of the publisher (Springer Science+Business Media, Inc., 233 Spring Street, New York, NY 10013, USA), except for brief excerpts in connection with reviews or scholarly analysis Use in connection with any form of information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now known or hereafter developed is forbidden The use in this publication of trade names, trademarks, service marks, and similar terms, even if they are not identified as such, is not to be taken as an expression of opinion as to whether or not they are subject to proprietary rights Mathematica, MathLink, and Math Source are registered trademarks of Wolfram Research, Inc Printed in the United States of America springeronline.com (HAM) To Carin, for her love, support, and encuragement Preface As physicists, mathematicians or engineers, we are all involved with mathematical calculations in our everyday work Most of the laborious, complicated, and time-consuming calculations have to be done over and over again if we want to check the validity of our assumptions and derive new phenomena from changing models Even in the age of computers, we often use paper and pencil to our calculations However, computer programs like Mathematica have revolutionized our working methods Mathematica not only supports popular numerical calculations but also enables us to exact analytical calculations by computer Once we know the analytical representations of physical phenomena, we are able to use Mathematica to create graphical representations of these relations Days of calculations by hand have shrunk to minutes by using Mathematica Results can be verified within a few seconds, a task that took hours if not days in the past The present text uses Mathematica as a tool to discuss and to solve examples from physics The intention of this book is to demonstrate the usefulness of Mathematica in everyday applications We will not give a complete description of its syntax but demonstrate by examples the use of its language In particular, we show how this modern tool is used to solve classical problems viii Preface This second edition of Mathematica in Theoretical Physics seeks to prevent the objectives and emphasis of the previous edition It is extended to include a full course in classical mechanics, new examples in quantum mechanics, and measurement methods for fractals In addition, there is an extension of the fractal's chapter by a fractional calculus The additional material and examples enlarged the text so much that we decided to divide the book in two volumes The first volume covers classical mechanics and nonlinear dynamics The second volume starts with electrodynamics, adds quantum mechanics and general relativity, and ends with fractals Because of the inclusion of new materials, it was necessary to restructure the text The main differences are concerned with the chapter on nonlinear dynamics This chapter discusses mainly classical field theory and, thus, it was appropriate to locate it in line with the classical mechanics chapter The text contains a large number of examples that are solvable using Mathematica The defined functions and packages are available on CD accompanying each of the two volumes The names of the files on the CD carry the names of their respective chapters Chapter comments on the basic properties of Mathematica using examples from different fields of physics Chapter demonstrates the use of Mathematica in a step-by-step procedure applied to mechanical problems Chapter contains a one-term lecture in mechanics It starts with the basic definitions, goes on with Newton's mechanics, discusses the Lagrange and Hamilton representation of mechanics, and ends with the rigid body motion We show how Mathematica is used to simplify our work and to support and derive solutions for specific problems In Chapter 3, we examine nonlinear phenomena of the Korteweg–de Vries equation We demonstrate that Mathematica is an appropriate tool to derive numerical and analytical solutions even for nonlinear equations of motion The second volume starts with Chapter 4, discussing problems of electrostatics and the motion of ions in an electromagnetic field We further introduce Mathematica functions that are closely related to the theoretical considerations of the selected problems In Chapter 5, we discuss problems of quantum mechanics We examine the dynamics of a free particle by the example of the time-dependent Schrödinger equation and study one-dimensional eigenvalue problems using the analytic and Preface ix numeric capabilities of Mathematica Problems of general relativity are discussed in Chapter Most standard books on Einstein's theory discuss the phenomena of general relativity by using approximations With Mathematica, general relativity effects like the shift of the perihelion can be tracked with precision Finally, the last chapter, Chapter 7, uses computer algebra to represent fractals and gives an introduction to the spatial renormalization theory In addition, we present the basics of fractional calculus approaching fractals from the analytic side This approach is supported by a package, FractionalCalculus, which is not included in this project The package is available by request from the author Exercises with which Mathematica can be used for modified applications Chapters 2–7 include at the end some exercises allowing the reader to carry out his own experiments with the book Acknowledgments Since the first printing of this text, many people made valuable contributions and gave excellent input Because the number of responses are so numerous, I give my thanks to all who contributed by remarks and enhancements to the text Concerning the historical pictures used in the text, I acknowledge the support of the http://www-gapdcs.st-and.ac.uk/~history/ webserver of the University of St Andrews, Scotland My special thanks go to Norbert Südland, who made the package FractionalCalculus available for this text I'm also indebted to Hans Kölsch and Virginia Lipscy, Springer-Verlag New York Physics editorial Finally, the author deeply appreciates the understanding and support of his wife, Carin, and daughter, Andrea, during the preparation of the book Ulm, Winter 2004 Gerd Baumann Contents Volume I Preface Introduction 1.1 Basics 1.1.1 1.1.2 1.1.3 1.1.4 1.1.5 1.1.6 Structure of Mathematica Interactive Use of Mathematica Symbolic Calculations Numerical Calculations Graphics Programming Classical Mechanics 2.1 Introduction 2.2 Mathematical Tools 2.2.1 Introduction 2.2.2 Coordinates 2.2.3 Coordinate Transformations and Matrices 2.2.4 Scalars 2.2.5 Vectors 2.2.6 Tensors 2.2.7 Vector Products 2.2.8 Derivatives 2.2.9 Integrals 2.2.10 Exercises vii 1 11 13 23 31 31 35 35 36 38 54 57 59 64 69 73 74 xii Contents 2.3 2.4 2.5 2.6 2.7 Kinematics 2.3.1 Introduction 2.3.2 Velocity 2.3.3 Acceleration 2.3.4 Kinematic Examples 2.3.5 Exercises Newtonian Mechanics 2.4.1 Introduction 2.4.2 Frame of Reference 2.4.3 Time 2.4.4 Mass 2.4.5 Newton's Laws 2.4.6 Forces in Nature 2.4.7 Conservation Laws 2.4.8 Application of Newton's Second Law 2.4.9 Exercises 2.4.10 Packages and Programs Central Forces 2.5.1 Introduction 2.5.2 Kepler's Laws 2.5.3 Central Field Motion 2.5.4 Two-Particle Collisons and Scattering 2.5.5 Exercises 2.5.6 Packages and Programs Calculus of Variations 2.6.1 Introduction 2.6.2 The Problem of Variations 2.6.3 Euler's Equation 2.6.4 Euler Operator 2.6.5 Algorithm Used in the Calculus of Variations 2.6.6 Euler Operator for q Dependent Variables 2.6.7 Euler Operator for q + p Dimensions 2.6.8 Variations with Constraints 2.6.9 Exercises 2.6.10 Packages and Programs Lagrange Dynamics 2.7.1 Introduction 2.7.2 Hamilton's Principle Hisorical Remarks 76 76 77 81 82 94 96 96 98 100 101 103 106 111 118 188 188 201 201 202 208 240 272 273 274 274 276 281 283 284 293 296 300 303 303 305 305 306 530 attracting set, 189 attracting sets, 189 average, 162 axial vector, 72 azimutal angle, 225 B backward scattering, 261 balance, 110 baseball, 95 beam, 269 beam intensity, 269 Bernoulli, 244, 291, 324 bi-soliton, 529 bifurcation, 149, 463 bifurcation diagram, 469 body centered coordinate, 478 body centered coordinates, 474 Boltzmann, 534 boundary conditions, 318 brachystochrone, 302 brachystochrone problem, 291 Brahe, 212 calculus of variation, 334 calculus of variations, 289 canonical equations, 428, 434 canonical momentum, 428 canonical transformation, 419, 424 canonical variables, 421 cartesian, 328 Index C cartesian coordinates, 332 Cartesian coordinates, 42, 68, 83 Cauchy, 386, 492 cenit angle, 225 center of mass, 220, 222, 256, 476 center of mass system, 256, 273, 486 center of mass velocity, 263 central field, 211 central field motion, 219 central force, 216, 223, 227, 269 central force problem, 219 central forces, 113, 211, 221 centrifugal force, 235, 238 centrifugal potential, 235 cgs system, 61 chaos, 189, 466, 511 chaotic, 115, 197 chaotic behavior, 194 chaotic dynamic, 460 chaotic entanglement, 195 chaotic motion, 189 Chaotic systems, 446 characteristic data, 519 characteristic frequency, 431 circular motion, 90 circular torus, 453 classical mechanics, 2, 34, 36 clock, 107 closed orbits, 232 cofactor, 48 Index collision, 255 column matrix, 45 complementary solution, 156 complete integrability, 435 completely integrable, 436 completely integrable equation, 520 complex behavior, 511 component, 41, 63 computer algebra, configuration space, 331 conic sections, 213, 244 conical sections, 244 conjungate momentum, 430 conservation law, 120, 264, 534 derivation, 534 conservation laws, 361 conservation of angular momentum, 362 conservation of energy, 534 conservation of mass, 534, 536 conservation of momentum, 362, 536 conservative, 127 conservative force field, 127 conserved quantity, 392, 402, 427 constraint, 382 constraint of non slip, 342 constraints, 316, 333 continuity equation, 534–535 continuous models, 511 continuous spectrum, 520 contour integral, 421 531 contravariant, 68 contravariant vector, 67–68 convex function, 376 coordinate, cyclic, 361 ignorable, 361 coordinate change, 419 coordinate system, 44 coordinate transformation, 76 coordinate transformations, 44 coordinates, 41 Coulomb scattering, 280 coupled pendulum, 347 Crank-Nicolson procedure, 539 critical damping, 149 critical phenomena, 470 critical point, 469 critically damped motion, 149 cross product, 72 curl, 80 current, 522, 534 cycle frequency , 434 cyclic, 361, 420 cyclic coordinate, 362 cyclic variable, 424 cyclic variables, 361 cycloid, 291 cylindrical coordinates, 419 D[], 11 damped harmonic oscillator, 144, 169 532 D damping constant, 190 damping factor, 160, 167 damping force, 144, 189 damping medium, 147 damping parameter, 144, 150 degrees of freedom, 189 density, 293, 298, 397 derivative, 11 derivatives, 40, 76 deviation moments, 477, 479 deVries, 511 difference method, 539 differentiable manifold, 407 differential equation, 13 differential scattering cross section, 269 differentiation rule, 401 diffusion, 314 Dirac Lagrangian, 311 Dirac's delta function, 515 direction, 63 direction cosine, 45 discrete eigenvalues, 521 discretization procedure, 540 dispersion, 517 dispersion relation, 514 dispersive, 514 distance, 104 division, dot product, 72 double pendulum, 416 Index drag force, 132 driven damped oscillator, 166 driven nonlinear oscillator, 188 driven oscillations, 155 driving force, 158, 189 driving frequency, 158–159 DSolve[], 129 DSolve[], 13 duration of oscillation, 175 dynamic, 189 dynamical principle, 327 dynamics, 83, 111 E Earth, 217 eccentricity, 244, 247 effective potential, 233, 235–236, 245 eigenfunction, 522 eigenvalue, 520–522 Einstein, 34 Einstein summation convention, 86 elastic collision, 255 electric field, 114 electromagnetic force, 117 electromagnetic forces, 252 Elements, 409 elevation, 99 ellipse, 142 ellipses, 213 elliptic fixpoints, 454 elliptic function, 180 Index elliptic integral, 180 elliptic integrals, 174, 231 EllipticK[], 180 elongation, 149 energy, 123, 142 energy loss, 148 energy of rotation, 235 energy resonance, 161, 163 equation of motion, 155, 228, 425 equilibrium position, 152 ergodic, 441 Euclidean plane, 294 Euler, 289, 376, 475, 489 Euler angles, 474, 487 Euler derivative, 289, 297, 310 Euler equation, 334 Euler Lagrange equations, 370 Euler method, 539 Euler operator, 299, 309, 312 Euler operator, 299 Euler procedure, 540 Euler theorem, 339 Euler-Lagrange equation, 345, 361, 375 Euler-Lagrange equations, 289, 334, 350, 384 Euler-Lagrange operator, 340 Euler's equation, 312 Euler's equations of motion, 487 Euler’s equation, 297 event, 107 evolution, 385 533 experimental facts, 104 exponentiation, external driving force, 155 external force, 108 external source, 155 F falling particle, 128 Feigenbaum, 468 Feigenbaum constant, 469 Ferma's principle, 324 Fermat, 324 Fermi, 511 field equation, 312 fields, 511 first integral, 332 first-order differential equations, 189 fixed interatomic distance, 474 fixed stars, 108 fixed system, 83 fixpoint, 453 flip chart movie, 360 flow, 436 flow field, 437 force, 111–112, 126, 331 attractive, 113 repulsive, 113 force center, 237, 273 force free symmetrical top, 492 force free top, 491 force moment, 491 534 forces, 63 forces in nature, 115 forward scattering, 261 Fourier transform, 514, 518 fractals, fractional, 470 frame of reference, 107 free body, 112 free oscillations, 155 free particle, 112 frequency, 137, 145, 181, 447 frequency of revolution, 235 friction, 155 frontend, functional, 292–293, 298, 308, 333–334 functional program, 30 fundamental Poisson brackets, 402 fundamental units, 61 G Galilean invariance, 536 Galilean transformation, 529 Galilei, 34 Galileo, 111 Gardner transformation, 537 Gauss, 326 general density, 534 general minimum principle, 325 generalized velocities, 375 generalized coordinates, 86, 89, 189, 332, 375 generalized coordinates, 43, 328 Index generalized momenta, 375, 434 generalized velocities, 328, 332 generating function, 422, 426, 429, 432, 439 generating functional, 292 generating functions, 421 Get[], 14 Giorgi system, 61 gold atoms, 283 golf play, 95 gradient, 78 gradient operator produc, 78 graphics, 16 gravitation, 211 gravitational constant, 110 gravitational field, 174, 219 gravitational force, 110, 132, 250 gravitational force, 115 gravitational mass, 111 gravitational masses, 110 gravity, 110, 115 Green's function, 164, 169–170 Green's method, 168 H hadronic force, 118 Hamilton, 34, 292, 327 Hamilton dynamics, 375 Hamilton equations, 439 Hamilton formulation, 375 Hamilton function, 378 Hamilton manifold, 414 Index Hamilton system, 442 Hamilton-Jacobi equation, 427, 430, 433 Hamilton-Jacobi theory, 428 Hamilton-Poisson manifold, 415 Hamiltonian, 382, 385, 387, 412, 416, 420, 423, 428–429, 431, 448, 450 Hamiltonian dynamics, 395 Hamiltonian formulation, 321 Hamiltonian phase space, 395 Hamilton's equation, 384, 403 Hamilton's equations, 386, 399 Hamilton's principle, 323, 327, 333, 339, 384, 388 Hamilton's principle, 332 HamiltonsEquation[], 386 hard spheres scattering, 278 harmonic oscillator, 136, 138, 140, 340, 431 heat, 147 Heisenberg's uncertainty, 34 Helmholtz, 127 help, 10 Henó, 450 Henó map, 450 Henon, 443 Hertz, 326 history, 107 homogeneity of space, 323 homogeneity of time, 323, 330 homogeneity relation, 363 homogeneous force field, 306 homogenous function, 338 homogenous functions, 339 535 Hooke's law, 137 Huberman, 470 Huygens, 235 hyperbolas, 213 hyperbolic fixpoint, 453 hyperlink, 10 hyperon, 36 I identity matrix, 48–49 impact parameter, 270, 273, 280 inclined plane, 341 incommensurable, 232 inelastic collision, 255 inertia, 66 inertia moments, 477 inertia tensor, 475, 477, 479, 489 inertial coordinates, 474 inertial frame, 108 inertial mass, 111 inertial reference frame, 108 infinite degree of freedom, 511 infinitesimal parameter, 364 infinitesimal rotation, 366 infinitesimal transformation, 364 inhomogeneous differential equation, 172 initial condition, 518 initial conditions, 140 input, input form, 12 input notation, 12 536 integrability, 375 integrable, 450 integral of motion, 428, 435 integral relation, 40 integrals, 80 integrals of motion, 435, 446 integration, 11 integro-differential equation, 514, 520 intensity, 269 interaction, 251 interaction laws, 252 interaction potential, 224, 235, 252, 521 interaction time, 255 interactive use, invariant, 72 invariants, 363, 419, 534, 536 Inverse[], 48 inverse matrix, 48 inverse scattering method, 514, 525 inverse scattering theory, 518 inverse scattering transform, 524 inversion, 167 involution, 435 isotropy of space, 330 iteration, 28 iterative mapping, 449 J Jacobi determinant, 457 Jacobi determinant , 449 Jacobi identity, 402 Index Jacobi matrix, 450 Jacobian, 379 Jacobian elliptic function, 186 Jacobi's identity, 411 JacobiSN[], 186 Josephson junction, 189 Joule, 127 Jupiter, 216 K KAM theorem, 442, 454 KdV, 511 KdV equation, 515 Kepler, 20, 212, 227 Kepler's laws, 213 kernel, 5, 10 keyboard short cuts, kinematics, 83 kinetic energy, 123, 175, 178, 225, 348 Kolmogorov, 442 Korteweg, 511 Korteweg-de Vries, 511 Kronecker delta symbol, 51 Kronecker's symbol, 477 Kruskal, 514, 540 L lab system, 266 label, laboratory system, 256, 261, 273 Lagrange, 34, 289, 318, 325 Lagrange function, 329 Index Lagrange density, 310, 335, 338–341, 357 Lagrange dynamics, 321, 375 Lagrange equations, 330–331, 344 Lagrange function, 307, 488 Lagrange multiplier, 318–319, 344–345 Lagrange's equation, 329 Lagrangian, 329–330, 363, 384, 419, 487, 489 Lagrangian formulation, 321 Lagrangien density, 350 l-calculus, 31 Landua, 330 Laplace, 330 Laplace equation, 314 Laplace transform, 13, 164, 169 Laplacian, 79 large wavelength, 515 latus rectum, 244 law of cosines, 267 laws of motion, 36 leap frog, 539 least action, 329 Legendre polynomial, 526 Legendre transform, 376 LegendreTransform[], 380 Leibniz, 324, 376 Leibniz's rule, 401, 406 length, 60 leptons, 119 Levi-Civita density, 73 Levi-Civita tensor, 489 lex prima, 111 537 lex secunda, 111 lex tertia, 111 libration, 175 Lie's symmetry analysis, 520 Lifshitz, 330 linear differential equations, 164 linear differential operator, 168 linear integral equation, 521 linear models, 511 linear momentum, 121 linear ordinary differential equation, 168 linear stability, 541 linearity, 401 Liouville, 400, 421 Liouville's theorem, 395, 400, 449 location of a particle, 83 log-log plot, 21 logistic function, 462 logistic map, 462, 468 Los Alamos, 514 Lyapunov exponent, 460, 466 M Mach, 105 magnetometers, 189 magnitude, 63 MANIAC, 514 manifolds and classes, 407 mapping, 449 mapping area, 449 mappings and Hamiltonians, 456 538 Marchenko equation, 514, 520, 524, 526–527, 539 Marchenko's integral equation, 524 mass, 60, 62, 104, 109–110, 112 mass center, 474 mass point, 83 material system, 37 Mathematica, mathematical approximation, 36 mathematical calculation, mathematical structure, 36 mathematical tools, 40 MathSource, 5, matrix, 45, 481 column, 45 inverse, 48 multiplication, 46 orthogonal, 51 square, 45 transposition, 47 Maupertius, 325 Maxwell’s equations, 312 mean distance, 216 mean distances, 245 measuring unit, 61 mechanics, 35 meson, 36 minimal principles, 323 minimum action, 325 minimum principle, 292 minor, 48 Index Miura, 514 Miura transformation, 536 mks system, 61 modulo, 191 modulus, 180 molecules, 114, 474 momentum, 112 Moser, 442 motion, 83, 109 motion of a ball, 96 motion of planets, 211 motion on a cylinder, 389 moving beat on a string, 381 moving coordinate, 515 moving frame, 43 multi-soliton, 520 multiplication, N N- particle system, 336 natural boundary conditions, 518 NDSolve[], 191 Neptune, 217 Newton, 34, 105, 213, 324 Newtonian mechanics, 104 Newtonian theory, 104 Newton's equation, 133, 334 Newton's equations, 323, 331 Newton's first law, 221 Newton's laws, 104, 111 Newton's second law, 221 Index Noether, 368 Noether theorem, 369 non integrability, 375 non-integrable, 450 nonholonomic, 333 nonlinear coupled chain, 514 nonlinear differential equations, 518 nonlinear dynamics, 511 nonlinear field equation, 511 nonlinear initial value problem, 520 nonlinear oscillation, 174 nonlinear partial differential equation, 519 nonlinearity, 517 Normal[], 182 normalization constant, 522 nucleon, 36 numerical calculation, 15 numerical integration, 15, 190 numerical solution, 190, 194 O object oriented programs, 31 observer, 107–108 operating system, optics, 323 options, 17 orbit, 231, 238 orbit potential, 234 orbits, 244 origin of time, 107 orthogonal matrix , 51 539 oscillatory motion, 136 output, overdamped motion, 150 P palettes, parabolas, 213 parabolic orbit, 96 parallelogram law, 114 parametric plot, 16 parametric representation, 19, 142 partial solution , 157 particle density, 534 particular solution, 156 Pasta, 511 path, 83, 306 pendula, 111 pendulum, 174, 179, 196 pendulum motion, 176 perihelia, 113 perihelion, 213, 246 period, 179, 181 period doubling, 468 periodic, 441, 446, 468 periodic regime, 470 periodic solution, 535 periodicity, 430 phase, 529 phase diagram, 140, 148 phase factor, 159, 161 phase plane, 140 540 phase portrait, 140 phase space, 177–178, 192, 195, 375, 400, 403, 419, 431, 435, 446, 451 phase space, 140 phase space volume, 422 phase transition, 470 phase velocity, 514–515 philosophy of mechanics, 107 physical approximation, 36 physical effect, 36 physical law, 104 physical laws, 104 physical theories, 36 pivot point, 189 planar pendulum, 188 planet motion, 238 planet movement, 211 planetary laws, 213 planetary motion, 233 platonic body, 214 plot, 16 Poincaré plane, 449, 452, 454 Poincaré section, 189, 193, 196–197, 458 Poincaré technique, 189 Poincaré-Hopf theorem, 436 point mass, 83 Poisson, 386 Poisson bracket, 400, 412, 414, 435 Poisson brackets, 400 Poisson manifold, 409, 412 PoissonBracket[], 404 Index polar axis, 225 polar coordinates, 42, 86 polynomial, 27 Pöschel, 525 Pöschel-Teller problem, 525 position, 83 position variable, 140 potential energy, 123, 126, 175, 331, 348 potential reconstruction, 521 power law, 231 power-law, 132 precession, 113 principal axes, 479 principal axis , 248 Principia, 111 principle of equivalence, 111 principle of least action, 329 principle of least constraint, 326 procedural function, 29 programming, 27 projectile, 95 Q quadratic equation, 10 quadrature, 175, 430 quantum mechanics, 2, 37, 520 quasi-periodic, 441, 446 Quit[], R radial equation, 228 radial oscillations, 232 Index radial velocity, 231 radial velocity , 233 random motion, 123 rank, 66, 68 rational number, 13 reaction, 113 recurrence, 232 reduced mass, 221 reference point, 83 reflection, 521 reflection coefficient, 521 reflection index, 523 reflection-less potential, 539 refraction, 324 regular dynamic, 511 regular motion, 189–190 relative coordinates, 219 relative motion, 107 relative velocity, 108 resonance, 161 resonance frequency, 161 rest, 112 restoring force, 136 revolutions, 178 rheonimic, 333 rigid body, 474, 478 rolling wheel, 318 rolling wheel, 341 rotating frame, 475 rotation, 474 rotation matrix, 49, 59 541 rotation symmetry, 269 rotations, 56 Rudnick, 470 Rudolphine table, 213 rule based program, 31 ruler, 107 Russel, 511 Rutherford scattering, 280 Rutherford's scattering formula, 282 S Sarturn, 217 scalar field, 40 scalar product, 71 scalars, 40, 60 scaling, 515 scaling exponent, 470 scaling law, 218, 470 scaling property, 469 scattering, 251 scattering angle, 260–261, 265, 270, 274 scattering cross section, 269, 271, 273, 283 scattering data, 520–521 scattering data , 519 scattering particles, 269 scattering potential, 520 scattering problem, 269, 520 scattering process, 519 Schrödinger's equation, 312 scleronomic, 333, 362 self-similar, 470 542 self-similar structure, 454 self-similarity, 454, 470 sensitivity, 189 separating variables, 179 separation, 428 separation ansatz, 526 separation of Hamiltonians, 433 separatrix, 178 shallow channels, 514 sliding beat, 387 sliding mass, 347 Snell's law, 324 solitary wave, 511 solitary waves, 514 soliton, 520, 525–526, 529 Solve[], 10 spectral characteristic, 524 spectral method, 539 spherical coordinates, 42, 225 spherical symmetry, 88, 224 spherical top, 490 square matrix, 45 standard form, 12 standard map, 458 standard package, 14 standard packages, StandardForm, 11 stationary characteristic, 521 stationary coordinate, 43 Steiner's theorem, 486–487 Index Stokes theorem, 421 strange attractor, 189, 196 strange entangled curve, 196 stroboscopic map, 193 stroboscopic snapshot, 189 strong nuclear force, 118 Sturm-Liouville problem, 518, 520–521 subtraction, sum, 12 super cyclic, 468 surface, 18 symbolic calculation, 10 symmetrical tensor, 477 symmetries, 361 symmetry, 123 symmetry analysis, 520 symmetry group, 149 symmetry line, 486 symmetry point, 486 symplectic matrix, 436 syntax, 1, T tangent map, 461 tangent representation, 460 target coordinates, 421 Taylor series, 12, 136 Taylor-Chiricov map, 458 Teller, 525 temperature, 60, 123 temporal change, 86 Index tensor, 66 rank, 66 tensors, 40 test function, 292–293 theoretical analysis, 36 theory of scattering, 520 thermal energy, 123 time, 60, 109 time, 104 time of revolution, 216 time-dependent potential, 521 top, spherical , 480 symmetric , 480 unsymmetrical, 480 topology, 436 tori, 446 torque, 122 torques, 63 torus, 437 total differential, 421, 538 total energy, 126, 138, 177, 233, 447 total kinetic energy, 475 total length, 294 traditional form, 12 trajectory, 430–431, 447 transformation matrix, 45 transformations, 40, 241 translation, 240, 474 translations, 121 translations in time, 362 543 transmission, 521 transmission coefficient, 521 transmission rate , 523 transposed matrix, 47, 49 transposition, 47 triangle addition law, 65 triangle law, 64 trigonometric function, 27 trigonometric functions, 9, 138 tunneling junction, 189 turning points, 231 twist map, 449 twist mapping, 450 two body problem, 211, 222, 251 two particle collision, 251 two-body forces, 114 two-dimensional oscillator system, 310 U Ulam, 511 underdamped motion, 145 uniform motion, 43, 112 uniformly accelerated, 43 units, 61 upper reversal point, 179 V vacuum, 132 variation, 308, 329 variational derivative, 314 variational principle, 323, 388 vector, 63–64, 67, 83 544 vector addition, 64 vector analysis, 14, 63 vector field, 40 vector product, 40, 71–72 vectors, 40 velocities, 63 velocity, 85 velocity, 104 velocity of sound, 133 Venus, 217 volume integration, 80 W water waves, 514 wave, 511 wave function, 520 weak nuclear force, 119 winding number, 448, 450 work, 123, 139 world-line, 107 Z Zabusky, 514, 540 Index .. .Mathematica? ? for Theoretical Physics ® Mathematica for Theoretical Physics Classical Mechanics and Nonlinear Dynamics Second Edition Gerd Baumann CD-ROM Included Gerd Baumann Department... English] Mathematica for theoretical physics / by Gerd Baumann. —2nd ed p cm Includes bibliographical references and index Contents: Classical mechanics and nonlinear dynamics — Electrodynamics,... The first volume covers classical mechanics and nonlinear dynamics The second volume starts with electrodynamics, adds quantum mechanics and general relativity, and ends with fractals Because of